The Kochen-Specker Theorem

The Kochen-Specker Theorem

Guillermo Morales Luna

Computer Science DepartmentCINVESTAV-IPN

[email protected]

Morales-Luna (CINVESTAV) Kochen-Specker Theorem 03/2007 1 / 27

Agenda

1 Observables and Uncertainty

2 C ∗-algebra of observables

3 Quantum Logic

4 Gleason’s Theorem

5 Kochen-Specker TheoremTheorem’s statementGeometrical motivationsSome implications

6 References


Agenda



3 Quantum Logic



6 References


Observables and Uncertainty

H: finite dimensional Hilbert space over C andSH: its unit sphere.A linear map U : H → H is self-adjoint if ∀x, y ∈ H 〈x|Uy〉 = 〈Ux|y〉, orequivalently: UH = U.A self-adjoint map is also called an observable.For an observable U : H → H there exists an ON basis of H consisting ofeigenvectors of U.If λ0, . . . , λk−1 are the eigenvalues of U and L0, . . . , Lk are thecorresponding eigenspaces the following implication holds:

x ∈ Lκ =⇒ U(x) = λκx.

Consequently, U is represented as

U =k−1∑κ=0

λκπLκ .


Since πLκ is an orthogonal projection, for each x ∈ H,〈x− πLκ(x)|πLκ(x)〉 = 0, thus

〈x|πLκ(x)〉 = 〈πLκ(x)|πLκ(x)〉 = ‖πLκ(x)‖2.

Extended measurement principle

For any observable U, when measuring an n-register x ∈ H, the output isan eigenvalue λκ and the current state will be the normalized projection

πLκ (x)‖πLκ (x)‖ . For each eigenvalue λκ, the probability that it is the output is

Pr(λκ) = 〈x|πLκ(x)〉. (1)

Evidently,∑k−1

κ=0 Pr(λκ) =∑k−1

κ=0 ‖πLκ(x)‖2 = ‖x‖2 = 1.


For any observable U, let (vj)j be an ON basis of H consisting ofeigenvectors of U: λj is the corresponding eigenvalue to vj .Any z ∈ SH, z =

∑i aivi , with

∑i |ai |2 = 1. And

〈z|Uz〉 =

⟨∑i

aivi |U

∑j

ajvj

⟩

=

⟨∑i

aivi |∑

j

ajλjvj

⟩=

∑i

λi |ai |2 = E (λi )

〈z|Uz〉 is the expected observed value of z under U.Standard deviation of U:

4U : H → R , x 7→ 4U(x) =√〈U2x|x〉 − 〈Ux|x〉2.


Let U1, U2 : H → H be two observables. Then ∀x ∈ H:

〈U2 ◦ U1x|x〉〈x|U2 ◦ U1x〉 = |〈U1x|U2x〉|2 = 〈U1 ◦ U2x|x〉〈x|U1 ◦ U2x〉,

and, from the Schwartz inequality, |〈U1x|U2x〉|2 ≤ ‖U1x‖2‖U2x‖2.

Robertson-Schrödinger Inequality

14|〈(U1 ◦ U2 − U2 ◦ U1)x|x〉|2 ≤ ‖U1x‖2‖U2x‖2.

The commutator of the two observables is [U1, U2] = U1 ◦ U2 − U2 ◦ U1.Observables U1, U2 are compatible if [U1, U2] = 0.

Heisenberg Principle of Uncertainty

|4U1(z)|2|4U2(z)|2 ≥14|〈z| [U1, U2] z〉|2 .


Agenda



3 Quantum Logic



6 References


C ∗-algebra of observables

Quantum system Σ: a collection of observables defined over a complexfinite dimensional Hilbert space H,AΣ = L(Σ): the C ∗-algebra generated by Σ.A linear functional f : AΣ → C is positive if

∀U ∈ AΣ : 〈f |U∗U〉 ≥ 0.

The identity map 1 is an unit in the C ∗-algebra AΣ.Each positive linear functional f : AΣ → C has norm ‖f ‖ = 〈f |1〉.A state over AΣ is a normalized positive linear functional:

f0, f1 states =⇒ ∀t ∈ [0, 1] : (1− t)f0 + tf1 state.

Each point x in the unit sphere SH can be identified as a state

x : U 7→ 〈x|Ux〉 : the expected value of x under U.


After Banach-Alaoglu Theorem: In the weak∗ topology, the unit ball iscompact: the set of states is compact in the weak∗ topology.For any U ∈ AΣ, the expected value of U at state z ∈ A∗Σ isEz(U) = 〈z |U〉.Spectrum of U: Λ(U) = {λ ∈ C|U − 1 is not invertible in AΣ}.The uncertainty is the variance of U at z :

Varz(U) = Ez (U − Ez(U)1)2 = Ez(U2)− Ez(U)2 ≥ 0.

Heisenberg Principle of Uncertainty is stated as follows:

Varz(A)Varz(B) ≥ 12|AB − BA| .


Agenda



3 Quantum Logic



6 References


Quantum Logic

Proposition: any observable with eigenvalues 0, 1, i.e. “yes” or “no”measures. Each proposition is an idempotent self-adjoint operator,A2 = A. Thus they are projections: each proposition corresponds to asubspace in the Hilbert space H.The “tautological” value 1 corresponds to the whole space H,“inconsistent” value to the null space {0}, the “conjunction” tointersection, or “meet”, and “disjunction” to “direct sum” or “linearunion”. “Negation” is thus orthogonal complementarity.If A1, A2 ∈ AΣ are propositions then

¬A1 = 1− AA1 ∧ A2 = lim

n→+∞(A1A2)

n

A1 ∨ A2 = ¬ (¬A1 ∧ ¬A2) = 1− limn→+∞

((1− A1) (1− A2))n


Spectral Theorem

Every self-adjoint operator can be split as the direct sum of the orthogonalprojections over its eigenspaces.

Every observable is the linear union of propositions that are mutuallycompatible and compatible with the given observable


Agenda



3 Quantum Logic



6 References


Gleason’s Theorem

Let H be a (real, complex, quaternion) Hilbert space and let

V(H) = {L < H| L is a closed linear space in H}.

V(H) is a complete orthomodular lattice.If {Lk}k∈K ⊂ V(H) is a collection of subspaces,

⊕k∈K Lκ is its supremum.

If {vι}ι∈I ⊂ H are vectors, L{vι}ι∈I ∈ V(H) is the span of the vectors.A measure is a map m : V(H) → R such that

m(H) = 1

{Lk}k∈K pairwise orthogonal =⇒ m

(⊕k∈K

Lκ

)=

∑k∈K

m (Lκ)

For instance, for any x ∈ SH, mx : L 7→ 〈x|πLx〉 is a measure.


Theorem (Gleason, 1957)Let H be a separable Hilbert space of dimension at least 3. For eachmeasure m there exists a Hermitian positive operator Tm such that

∀L ∈ V(H) : m(L) = Tr(TmπL).

Besides, the map m ↔ Tm is a bĳection among measures and Hermitianpositive operators.

Since SH is connected and the map L 7→ Tr(TmπL) is continuous (in atopology well defined in V(H)) then, from Gleason’s Theorem, there areno (“yes”,“no”)-measures in the space of propositions.Since SH is weak∗-compact, there is finite collection of subspaces in V(H)in which no non-trivial two-valued measure can be defined. A constructionof such an example is given by Kochen-Specker Theorem.


Agenda



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6 References


Kochen-Specker Theorem

H: a separable complex Hilbert space. Σ: collection of observables.µ : Σ → R: a map assigning to an observable a definite value.

Property (Value definiteness (VD))

At any time µ is a total map: The values of the observables are welldetermined at any time.

Property

The map µ fulfills the following two rules:Sum rule If U0, U1, U2 ∈ Σ are compatible then:

[U2 = U0 + U1 =⇒ µ(U2) = µ(U0) + µ(U1)].Product rule If U0, U1, U2 ∈ Σ are compatible then:

[U2 = U0U1 =⇒ µ(U2) = µ(U0)µ(U1)].


If µ is a “yes”-“no” assignment and (Li )i is a finite set of propositions(their projections πli have eigenvalues 0,1), the sum rule implies:

1 = µ(πL

i Li

)=∑

i

µ (πLi ) .

Thus necessarily:(VC)

[µ (πLi ) = 1 =⇒ µ

(πLj

)= 0 ∀j 6= i

].

The product rule implies:(VE) [L < L0 ⊕ L1 =⇒ µ (πL) µ (πL0⊕L1) = µ (πL)] .


Paint in color red those propositions s.t. µ (πLi ) = 1 and in green thosepropositions s.t. µ (πLi ) = 0.The sum rule and condition (VC) imply that in any finite ON system ofvectors exactly one vector is red and the others are green.Condition (VE) implies that any proposition in the span of two greenpropositions should be colored green.

Theorem (Kochen-Specker (KSThm, 1967))Let H be a separable complex Hilbert space of dimension at least 3. Thenthere exists a collection of observables Σ such that Property 1 andProperty 2 cannot hold simultaneously.

Theorem (Geometrical tridimensional real Kochen-Specker –)

In R3 there exists a collection of rays Σ such that the following cannothold simultaneously for any “green”-“red” coloring:

1 In any triplet of orthogonal rays exactly one is colored red.2 Any ray lying in the span of two green rays is colored green.


Geometrical motivation

x = (cos(θ) cos(φ), sin(θ) cos(φ), sin(φ)): a point in the unit sphere of R3.θ: the longitude. φ: the latitude.xE = (− sin(θ), cos(θ), 0) :A point in the equator orthogonal to x (itslongitude is that of x shifted by an angle of π/2 radians).x⊥ = x× xE = (− cos(θ) sin(φ),− sin(θ) sin(φ), cos(φ)) .{

x, xE , x⊥}

is a positively oriented ON basis of R3.Let us suppose x is in the northern hemisphere of the unit sphere and it isnot the North Pole. By changing the direction of xE if necessary, we mayassume that x⊥ lies also in the northern hemisphere.The circle with center at the origin passing over x and xE is the descentcircle of x.


Mapamundi centered at San Sebastián


A descent sequence {xi}ki=0 is a sequence built as follows: x0 is a point in

the northern hemisphere which is not the North Pole. For any i ≥ 0, xi+1is a point chosen on the descent circle of xi (to the south of xi ).

LemmaGiven two points in the northern hemisphere with different latitude, thereis a descent sequence beginning with the northern point and ending in thesouthern point.

The proof is direct by using the projection x 7→ 1x3

x mapping each point inthe northern hemisphere into the crossing of its ray with the plane parallelto the x , y -plane passing by the North Pole. Parallel circles correspondingto same latitudes are mapped into concentric circles, and descent circlesinto tangents to concentric circles.


Sketch of Thm 3

Let initially Σ = {e0, e1, e2} be the collection of the three canonical basicvectors. Let us assume that the North Pole e2 = (0, 0, 1) is colored red.Let x be a vector of latitude 1

3π. Then xE is in the equator, hence coloredgreen, and x⊥ has latitude 1

6π. It follows that x and x⊥ have oppositecolors.If x is green then on one side x⊥ is red and on the other any ray in thedescent circle of x is green. Thus any ray reached by a descent sequencefrom x should be green, in particular x⊥ and this is a contradiction.Otherwise, any ray at an angle of 1

6π from the North Pole, shall be red, asis that pole. Indeed any ray in the cone of angle 1

6π of a red ray is red.Then there is an arc of three red rays from the North Pole to the Equator,another arc of three red rays along the Equator, and another arc of threered rays from the Equator to the North Pole. The three corners in thiscircuit are red, but they form an orthogonal system. This is acontradiction.


Some implications

Roughly speaking, KSThm implies that Quantum Mechanics (QM) is notconsistent with the following two properties:Value definiteness (VD) All observables have definite values at all times.Non Contextuality (NC) If an observable aquires a value, it does so

independently of any measurement context.In symbols:

KSThm: QM 6|= VD + NC

Consequently, acceptance of QM entails a rejection of either VD or NC.VD is the origin of any hidden variables programme.NC is the motivation of the notion of realism.It is an important problem how to come up with a version of QMcontaining VD but not NC.


Agenda



3 Quantum Logic



6 References


References

R.D. Gill, Discrete Quantum Systems, lecture notes, University Utrecht,http://www.math.uu.nl/people/gill/ , 1995.

R.D. Gill, Notes on Hidden Variables, lecture notes, University Utrecht,http://www.math.uu.nl/people/gill/ , 1995.

R. D. Gill and M. S. Keane, A geometric proof of the Kochen-Specker no-gotheorem J. Phys. A: Math. Gen., 29, L289-L291, 1996.

C. Held, The Kochen-Specker Theorem, Stanford Encyclopedia ofPhilosophy, http://plato.stanford.edu/entries/kochen-specker/ ,2006.


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The Kochen-Specker Theorem